77

7

9 35 2 1 4108 6 R 4

()

9

Y X9

A<

R

A < X7 8

1- 0 () ) +

.2 - )4 6 ( =

•••••••••• !"

# $

• 2 1 6 5/

• / 1• 3/4 /

2

確率

変数

€

a

€

b

€

P a ≤ x ≤ b( )

•

€

x

••••

••••

•

•

•

• ! "# $# (! % &' &

• % = ")$) + "'$' + ⋯+ ",$,( )

• &' = ") − % '$) + "' − % '$' + ⋯+ ", − % '$,• &

)

! ") "' ". ",$ $) $' $. $,

1!

/ "

01/

1

! " = 4

)4(

• /56 ! = 1× ⁄1 6 + 2× ⁄1 6 +⋯+ 6× ⁄1 6 = 3.5• 2-. = 1 − 3.5 .× ⁄1 6 + 0. − ! .× ⁄1 6 +⋯+ 01 − ! .×⁄1 6 ≅ 2.9• 3 1- ≅ 1.7

• ! "# $ $% = '$ + )$% * $% + $%

• * $% = ∑-./0 '1- + ) 2- = ' ∑-./0 1-2- + ) ∑-./0 2- = '! + )• + $% = ∑-./0 '1- + ) − '! + ) #2- = '# ∑-./0 1- − ! #2- = '#"#

! "#, "%,⋯ , "',⋯"'

() ! = "' = + "'

+ "' ≥ 0,.'/#

0+ "' = 1

!

• ! " X

" ! = %&'(

)*&+ *&

= %&'(

)*&,- ! = *&

•, ! E X ! $ , %

E % = %

E % ! = %' !

E ! + $ = ' ! + ' $

• ! Var ! %&

Var ! = ( ! − ( ! * = +,-.

/0, − 1 *2 0,

%& = Var !

1 0,

• ! "#$ !

"#$ % = 0

"#$ ! + % = "#$ !

"#$ %! = %)"#$ !

• ,1

• 0

• 2 1 1 / 11 1 1 1

2

• ) ( (!

• ( "#" "#$( "#" ×! "#$ × 1 − !

• (( ( (#( ×!(

(#( ×!( (#" ×! 1 − ! (#( × 1 − ! (•

!"! ×$ !"% × 1 − $

("( ×$( ("% × 1 − $ (("! ×$ 1 − $

)") ×$) )"( ×$( 1 − $ )"! ×$ 1 − $ ( )"% × 1 − $ )

• 0 ! 1 " 1# 1

0• 1 11 0

$%!&' !, # = *+, - #, - 1 − # *0, ≈ #, - 1 − # *0,

•

• ! = #$• %& = #$ 1 − $

) 0 1

1 − $ $

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.01

0.02

0.03

0.04

p

L(p)

0 321) (

050

100

150

200

250

得点

チーム数

0 1 2 3 4 5 6 7 8 9 10

.

0 . .

J.League Data Siteより作成

•

•!• " # ! = "#

%& ! = !' ( )*# −!

*!

0 5 10 15 20

0.0

0.1

0.2

0.3

x

確率

Po(1)

Po(3)

Po(5)

Po(10)

0 5 10 15 20

0.0

0.1

0.2

0.3

x

確率

Po(1)

Po(3)

Po(5)

Po(10)

発生件数＝離散的な値

!" # = #% & '() −#

(!

7 6425( ** 3/10 **! ≑ *( ** /10 **! ≑( .

( ( **( /10 **! ( ≑) ( **) /10 **! ) ≑-(* ( *** /10 **! * ≑(

( ** /10 **! ≑-( ** /10 **! ≑(

- ( ** /10 **! ≑

132

227

154

66

23 6 4 0

146

209

150

72

26 7 2 0 0

50

100

150

200

250

0 1 2 3 4 5 6 7

実際の得点 ポアソン分布に従う得点

試合×チーム数

得点

Po(1.44)

••

! " = $%&'(, " ≥ 0

• , - . -, - = 1$

. - = 1$0

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

• ! = # − 1, ' − ! = ( − 1 )*'+, ', -) ', -

)*'+, ', - ≈ -/ 0 1 − - 12/

= # − 1 ! ( − 1 !# + ( − 1 !

) ', - = 5 0 -627 0 1 − - 8270 < - < 1, 0 < #, 0 < (

• 5

• ! ", $ % &'

% = )) + +

&' = )+) + + ' ) + + + 1

0.0 0.2 0.4 0.6 0.8 1.00.6

0.8

1.0

1.2

1.4

x

y

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

x

y

! 3, 2! 3, 2

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

2.5

3.0

x

dbet

a(x,

0.5

, 0.5

)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

x

dbet

a(x,

5, 5

)

! 0.5, 0.5 ! 5, 5

•••••

•

•

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

x

dbet

a(x,

10,

15)

プロジェクト終了期間

•

!"#$% #, ' ≈ ') * 1 − ' -./= 1 − 1 ! 3 − 1 !1 + 3 − 1 != Γ 1 Γ 3Γ 1 + 3

• Γ 1

• Γ "

Γ " = $%

&'()*+), -'

" > 0

• Γ "

Γ " = " − 1 !Γ " + 1 = "Γ "Γ ⁄1 2 = 6

• ! "

! " = $% &, ( = ()

Γ & ")+,-+./, " ≥ 0

• 2 3 2 4 2

3 2 = &(

3 2 = &(5

Γ(0.5) Γ(10)

0.0 0.2 0.4 0.6 0.8 1.0

0e+00

2e-07

4e-07

6e-07

8e-07

1e-06

x

0.0 0.2 0.4 0.6 0.8 1.0

05

1015

x

Γ 0.5 Γ 10

• ! "# $% !, "#

% !, "# = ( $ = 12+"#

,$- − $ − !#

2"#

•

/ 0 = !1 0 = "#

確率

変数２ ３ ４ ５ ６ ７ ８ ９ 10 11 12

11

21

12

22

31

13

32

23

41

14

33

42

24

51

15

43

34

52

25

61

16

44

53

35

26

62

54

45

36

36

55

46

46

65

56

66

• !" = ⁄! − & '

• " & = 0 ') = 1+ 0,1

• + 0,1 - "

+ 0,1 = - " = 12/ 0!1 −")2

•

• , 1 ( ))-

Binom n, p( ) ≈ pn ⋅ 1− p( )n−s

N np,np 1− p( )( )

=1

2πnp 1− p( )exp −

x − np( )2

2np 1− p( )

"

#$$

%

&''

!"•1 #$ % &" #$~( %, &"

*$ = ,-./0*$~( 0,1• *$" !" !" 1

*$" =#$ − %&

"~!" 1

• 4 = ∑$678 *$" 9 !" !" 94 =:

$67

8*$" ~!" 9

0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

x

dchi

sq(x

, 1)

!"!"

!" !" 1

!" 3!" 5

!" 10!" 20

!"• #0!" !" # 0 $ %; # 0 ≤ %0

% < 00

$ %; # = 12-"Γ #2

/01"%

-"02

• Γ 3• #0!" 0 () # 2#

!"• #$ ⁄& 2 , 2 & !" !" &

#$ &2 , 2 = !" &

• * = +" #$ ⁄1 2 , ⁄1 2

#$ 12 ,12 =

12-+"

.*/ −+"

2